Purple sweet potatoes make for a dramatic presentation, but regular orange varieties also work well.
Preheat oven to 425°F. Place sweet potatoes on a large rimmed baking sheet. Bake in preheated oven until tender, about 40 minutes.
Meanwhile, heat 1 tablespoon of the oil in a medium saucepan over medium-high. Add onion and turmeric; cook, stirring often, until softened, about 3 minutes. Add stock and quinoa; let come to a boil. Reduce heat to medium; cover and simmer, stirring occasionally, until quinoa is tender and liquid is almost absorbed, about 10 minutes. Stir in kale, pepper, and 1/2 teaspoon of the salt; cook, covered, stirring occasionally, until kale is wilted, about 5 minutes. Remove from heat.
Cut a lengthwise slit about 1 1/2 inches deep into each potato. Push in potato ends to slightly open, and gently fluff flesh using a fork. Sprinkle potatoes evenly with remaining 1/2 teaspoon salt. Spoon about 1/2 cup quinoa mixture into each potato; dollop evenly with yogurt, and sprinkle with pomegranate arils. Drizzle evenly with remaining 2 tablespoons oil.